We study the evolution of a proto-giant-planet envelope contracting self-gravitationally and examine the dynamical stability of the envelope. There are two opposing studies on how the envelope evolves. P. Bodenheimer and J. B. Pollack (1986, Icarus67, 391-408) argued that the protoplanet contracts quasi-statically and becomes a giant planet with a massive envelope. In contrast, G. Wuchterl (1991b, Icarus91, 53-64), under a hydrodynamic assumption, showed that the mass of the planetary envelope cannot immediately grow to the present mass of Jupiter. This is because a hydrodynamical wave is excited by the κ-mechanism and ejects envelope mass immediately after the core mass increases beyond the static critical mass. We adopt a quasi-static numerical code which is quite independent of Bodenheimer and Pollack (1986) and compute the structure and evolution of an envelope of a protoplanet in Jupiter's region, using the same opacity data as Bodenheimer and Pollack (1986). In addition, as a self-consistency check, we examine the dynamical stability of the envelope by means of a quasi-adiabatic oscillation analysis in each time step of the evolution. Under nominal solar nebular conditions and a quasi-static assumption, we pursue the evolution to the stage where the mass of the planet becomes as large as the present Jupiter. At that time, we find that the envelope contracts quasi-statically during the entire evolution and giant planets having massive envelopes can be formed; hence our results are quite similar to those obtained by Bodenheimer and Pollack (1986).